| import Mathlib |
| import OAI.Geometry.Cannon.Energy.RawFunction |
|
|
| namespace OAI |
|
|
| section |
| section |
| section |
| noncomputable section |
| section |
| section |
| section |
| section |
| section |
| open Set Topology |
| open Set Topology Filter Metric unitInterval |
| open scoped InnerProductSpace |
| open Set unitInterval |
| open Set Metric Topology |
| open Set Topology unitInterval |
| open Set Filter Topology |
|
|
| namespace CannonChart |
|
|
| open Set Topology Metric |
|
|
| variable {X Y : Type*} |
|
|
| def extend (e : X → Y) (f : X → ℝ) : Y → ℝ := Function.extend e f (fun _ => 0) |
|
|
| lemma extend_apply (e : X → Y) (he : Function.Injective e) (f : X → ℝ) (x : X) : |
| extend e f (e x) = f x := he.extend_apply f (fun _ => 0) x |
|
|
| lemma extend_image (e : X → Y) (he : Function.Injective e) (f : X → ℝ) |
| (A : Set Y) (hA : A ⊆ range e) : |
| extend e f '' A = f '' (e ⁻¹' A) := by |
| ext t |
| constructor |
| · rintro ⟨y,hy,rfl⟩ |
| obtain ⟨x,rfl⟩ := hA hy |
| exact ⟨x,hy,(extend_apply e he f x).symm⟩ |
| · rintro ⟨x,hx,rfl⟩ |
| exact ⟨e x,hx,extend_apply e he f x⟩ |
|
|
| lemma continuousOn_extend [TopologicalSpace X] [TopologicalSpace Y] (e : X → Y) (he : IsEmbedding e) (f : X → ℝ) |
| (hf : Continuous f) : ContinuousOn (extend e f) (range e) := by |
| rw [continuousOn_iff_continuous_domRestrict] |
| have hc := hf.comp he.toHomeomorph.symm.continuous |
| convert hc using 1 |
| funext y |
| obtain ⟨x,hx⟩ := y.property |
| have hy : y = he.toHomeomorph x := Subtype.ext hx.symm |
| subst y |
| simp only [Set.domRestrict, Function.comp_apply, Homeomorph.symm_apply_apply] |
| exact extend_apply e he.injective f x |
|
|
| end CannonChart |
|
|
| namespace CannonApprox |
|
|
| open CannonGrid CannonLevel CannonChart Metric |
|
|
| theorem exists_chart_lateral_diameter {X : Type*} [MetricSpace X] |
| (e : CannonGrid.Square → X) (he : Topology.IsEmbedding e) : |
| ∃ d₁ : ℝ, 0 < d₁ ∧ ∀ A : Set CannonGrid.Square, IsCompact A → |
| (∃ x ∈ A, x.1 = 0) → (∃ x ∈ A, x.1 = 1) → d₁ ≤ diam (e '' A) := by |
| let l : Set CannonGrid.Square := {z | z.1 = 0} |
| let r : Set CannonGrid.Square := {z | z.1 = 1} |
| have hl : IsCompact l := (isClosed_eq continuous_fst continuous_const).isCompact |
| have hr : IsCompact r := (isClosed_eq continuous_fst continuous_const).isCompact |
| have hdis : Disjoint (e '' l) (e '' r) := by |
| apply Set.disjoint_left.mpr |
| rintro z ⟨x,hx,hxz⟩ ⟨y,hy,hyz⟩ |
| have hxy := he.injective (hxz.trans hyz.symm) |
| have h01 : (0 : unitInterval) = 1 := hx.symm.trans ((congrArg Prod.fst hxy).trans hy) |
| exact zero_ne_one h01 |
| obtain ⟨d,hd,hsep⟩ := Metric.exists_pos_forall_lt_edist |
| (hl.image he.continuous) (hr.image he.continuous).isClosed hdis |
| refine ⟨d,by exact_mod_cast hd,?_⟩ |
| intro A hA hAl hAr |
| obtain ⟨x,hx,hxl⟩ := hAl |
| obtain ⟨y,hy,hyr⟩ := hAr |
| have hh := hsep (e x) ⟨x,hxl,rfl⟩ (e y) ⟨y,hyr,rfl⟩ |
| rw [edist_nndist] at hh |
| have hdist : (d : ℝ) ≤ dist (e x) (e y) := by exact_mod_cast hh.le |
| exact hdist.trans (dist_le_diam_of_mem (hA.image he.continuous).isBounded |
| (mem_image_of_mem e hx) (mem_image_of_mem e hy)) |
|
|
| theorem scalar_ambient_level {X : Type*} [MetricSpace X] |
| (e : CannonGrid.Square → X) (he : Topology.IsEmbedding e) |
| {ι : Type*} [Fintype ι] (C : ι → Set CannonGrid.Square) (ρ : ι → ℝ) |
| (hρ : ∀ i, 0 ≤ ρ i) (hadm : Admissible C ρ) (n : ℕ) (hn : 0 < n) |
| (d₁ : ℝ) (hd₁ : ∀ A : Set CannonGrid.Square, IsCompact A → |
| (∃ x ∈ A, x.1 = 0) → (∃ x ∈ A, x.1 = 1) → d₁ ≤ diam (e '' A)) |
| (t : ℝ) (ht : t ∈ Ioo (0:ℝ) 1) : |
| ∃ A : Set X, IsCompact A ∧ IsConnected A ∧ A ⊆ e '' centralRectangle ∧ |
| d₁ ≤ diam A ∧ (∀ z ∈ A, extend e (scalar C ρ n hn) z = t) ∧ |
| (∀ {a b : CannonGrid.Square} (p : Path (e a) (e b)), |
| (∀ s, p s ∈ range e) → scalar C ρ n hn a < t → t < scalar C ρ n hn b → |
| ∃ s, p s ∈ A) := by |
| obtain ⟨A,hA,hcon,hcentral,hf,hl,hr,hhit⟩ := scalar_level C ρ hρ hadm n hn ht |
| refine ⟨e '' A,hA.image he.continuous,hcon.image e he.continuous.continuousOn, |
| image_mono hcentral,hd₁ A hA hl hr,?_,?_⟩ |
| · rintro z ⟨x,hx,rfl⟩ |
| rw [extend_apply e he.injective] |
| exact hf x hx |
| · intro a b p hp ha hb |
| obtain ⟨s,hs⟩ := hhit (liftPath e he p hp) ha hb |
| refine ⟨s,?_⟩ |
| rw [← liftPath_apply e he p hp s] |
| exact mem_image_of_mem e hs |
|
|
| end CannonApprox |
|
|
| open Set MeasureTheory |
| open scoped BigOperators |
| open Set unitInterval |
|
|
| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
|
|
| end OAI |
|
|
